Question

Дам 30 балов!!! Осевое сечение конуса правильный треугольник со стороной 10 см.Найдите радиус основания и высоту конуса.

Answer 1

Finding the Radius and Height of a Cone

To find the radius and height of a cone, given that the cross-section is an equilateral triangle with a side length of 10 cm, we can use the properties of an equilateral triangle and the formula for the volume of a cone.

Let's start by finding the radius of the cone.

Step 1: Finding the Radius

In an equilateral triangle, all sides are equal. Since the side length of the equilateral triangle is given as 10 cm, each side of the triangle is also 10 cm.

In a cone, the radius of the base is equal to the length of the side of the equilateral triangle. Therefore, the radius of the base of the cone is 10 cm.

Step 2: Finding the Height

To find the height of the cone, we can use the formula for the volume of a cone:

V = (1/3) * π * r^2 * h

where V is the volume, r is the radius, and h is the height of the cone.

Since the cross-section of the cone is an equilateral triangle, the height of the cone is equal to the height of the equilateral triangle.

In an equilateral triangle, the height can be found using the formula:

h = (sqrt(3) / 2) * s

where h is the height and s is the side length of the equilateral triangle.

Substituting the given side length of 10 cm into the formula, we get:

h = (sqrt(3) / 2) * 10 cm

Simplifying the equation, we find that the height of the cone is approximately 8.66 cm.

Therefore, the radius of the base of the cone is 10 cm and the height of the cone is approximately 8.66 cm.

Please note that the calculations provided are based on the assumption that the cross-section of the cone is a regular equilateral triangle. If there are any additional specifications or constraints, please let me know and I will be happy to assist you further.